Double and dual numbers

Hypercomplex numbers of the form $ a + be $, where $ a $ and $ b $ are real numbers, and where the double numbers satisfy the relation $ e ^ {2} = 1 $, while the dual numbers satisfy the relation $ e ^ {2} = 0 $( cf. Hypercomplex number). Addition of double and dual numbers is defined by

$$ ( a _ {1} + b _ {1} e) + ( a _ {2} + b _ {2} e) = \ ( a _ {1} + a _ {2} ) + ( b _ {1} + b _ {2} ) e. $$

Multiplication of double numbers is defined by

$$ ( a _ {1} + b _ {1} e) ( a _ {2} + b _ {2} e ) = \ ( a _ {1} a _ {2} + b _ {1} b _ {2} ) + ( a _ {1} b _ {2} + a _ {2} b _ {1} ) e , $$

and that of dual numbers by

$$ ( a _ {1} + b _ {1} e )( a _ {2} + b _ {2} e ) = a _ {1} a _ {2} + ( a _ {1} b _ {2} + a _ {2} b _ {1} ) e . $$

Complex numbers, double numbers and dual numbers are also called complex numbers of hyperbolic, elliptic and parabolic types, respectively. These numbers are sometimes used to represent motions in the three-dimensional spaces of Lobachevskii, Riemann and Euclid (see, for instance, Helical calculus).

Both double and dual numbers form two-dimensional (with base 1 and $ e $) associative-commutative algebras over the field of real numbers. As distinct from the field of complex numbers, these algebras comprise zero divisors, all these having the form $ a \pm ae $ in the algebra of double numbers. The algebra of double numbers may be split into a direct sum of two real number fields. Hence yet another name for double numbers — splitting complex numbers. Double numbers have yet another appellation — paracomplex numbers. The algebra of dual numbers is considered not only over the field $ \mathbf R $ of real numbers, but also over an arbitrary field or commutative ring. Let $ A $ be a commutative ring and let $ M $ be an $ A $- module. The direct sum of $ A $- modules $ A \oplus M $ equipped with the multiplication

$$ ( a , m ) ( a ^ \prime , m ^ \prime ) = ( aa ^ \prime , am ^ \prime + a ^ \prime m ) $$

is a commutative $ A $- algebra and is denoted by $ I _ {A} ( M) $. It is known as the algebra of dual numbers with respect to the module $ M $. The $ A $- module $ M $ is identical with the ideal of the algebra $ I _ {A} ( M) $ which is the kernel of the augmentation homomorphism

$$ \epsilon : I _ {A} ( M) \rightarrow A \ ( ( a , m ) \rightarrow a ) . $$

The square $ M ^ {2} $ of this ideal is zero, while $ I _ {A} ( M) / M \simeq A $. If $ A $ is a regular ring the converse is also true: If $ B $ is an $ A $- algebra and $ M $ is an ideal in $ B $ such that $ M ^ {2} = 0 $ and $ B/M \simeq A $, then $ B \simeq I _ {A} ( M) $, where $ M $ is regarded as an $ A $- module [4].

If $ M = A $, the algebra $ I _ {A} ( M) $( then denoted by $ I _ {A} $) is isomorphic to the quotient algebra of the algebra of polynomials $ A( T) $ by the ideal $ T ^ { 2 } $. Many properties of an $ A $- module may be formulated as properties of the algebra $ I _ {A} ( M) $; as a result, many problems on $ A $- modules can be reduced to corresponding problems in the theory of rings [2].

Let $ B $ be an arbitrary $ A $- algebra, let $ \phi : B \rightarrow A $ be a homomorphism and let $ \partial : B \rightarrow M $ be a derivation (cf. Derivation in a ring) of $ B $ with values in the $ A $- module $ M $, regarded as a $ B $- module with respect to the homomorphism $ \phi $. The mapping $ \overline \partial \; : B \rightarrow I _ {A} ( M) $( $ b \rightarrow ( \phi ( b), \partial ( b)) $) will then be a homomorphism of $ A $- algebras. Conversely, for any homomorphism of $ A $- algebras $ f : B \rightarrow I _ {A} ( M) $ the composition $ \epsilon ^ \prime \circ f : B \rightarrow M $, where $ \epsilon ^ \prime : I _ {A} ( M) \rightarrow M $ is the projection of $ I _ {A} ( M) $ onto $ M $, is an $ A $- derivation of $ B $ with values in $ M $, regarded as a $ B $- module with respect to the homomorphism $ \epsilon \circ f : B \rightarrow A $. This property of double and dual numbers is utilized for the description of the tangent space to an arbitrary functor in the category of schemes [1], [3].

References

[1]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)
[2]	R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975)
[3]	M. Demazure, A. Grothendieck, "Schémas en groupes I" , Lect. notes in math. , 151–153 , Springer (1970)
[4]	S. Lichtenbaum, M. Schlessinger, "The cotangent complex of a morphism" Trans. Amer. Math. Soc. , 128 : 1 (1967) pp. 41–70

Comments

An old-fashioned term for an associative algebra $ A $ with unit element over $ \mathbf R $ is system of hypercomplex numbers, and an element of $ A $ is then called a hypercomplex number. There are (up to isomorphism) precisely three of these algebras of dimension 2: the complex numbers, the dual numbers and the double numbers.

References

[a1]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970)